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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), by Cao Zexian (Institute of Physics, Chinese Academy of Sciences)

Poincare is a great mathematician, physicist, philosopher and engineer in contemporary France, known as the last person who knows everything, and his knowledge has reached the depth of phenomenal level. Poincare was a self-contained scientist who dominated the mathematics and physics of his time and made great contributions to all the fields in which he devoted himself. His contributions to the theory of relativity and quantum mechanics were foundational and decisive. Poincare is also a philosopher, in fact, the use of contingency has a universal guiding significance for scientific practice, and his elegant words of popularizing science will continue to influence the world.

Mediocre people pay attention to extraordinary things, great people pay attention to ordinary things.

-- Pascal

1. There is a classification of introduction physics according to the number of research objects, including monomer problem, two-body problem, few-body problem and multi-body problem. In recent years, many people have added the concept of three-body to their chat vocabulary. The three-body problem (three-body problem) was at first a particularly natural astrophysical question: is the gravitational interaction of the sun-earth-moon stable? This is a standard alarmist worry. The dynamics of the three-body has no general solution in closed form, and it will show chaotic behavior (chaotic behavior) for the general initial conditions. Chaos has since become an important interdisciplinary concept. This concept comes from the work of the great French mathematician Poincare. In 1890, Poincare provided systematic ideas and mathematical techniques for the solution of three-body dynamics in a 270-page paper, and made concepts such as chaos social.

When it comes to mathematicians, it is very interesting to say that the mathematician criterion has something to do with Poincare Lemma. If a person is fast asleep, you kick him up and ask, "what is the Poincare Lemma?" If you can't answer, you're definitely not a mathematician. Many people will be surprised that Poincar é's lemma has such a high status that it can be used as a criterion for mathematicians. Poincar é's Lemma talks about the zero-tone property of differential forms on an open unit ball. If U is the kickoff of a Rn space and Ek (U) is a differential k-form (differential k-form) space on U, then for k ≥ 1, there is a linear transformation.

Figure 1. Poincare in the study Poincare's famous mathematical achievements include:

1. Automorphic functions, uniformization (automorphic function, uniformization)

2. The qualitative theory of differential equations (qualitative theory of differential equations)

3. Bifurcation theory (bifurcation theory)

4. Asymptotic expansions, normal forms (Asymptotic expansion, Norm)

5. Dynamical systems, integrability (dynamic system, integrability)

6. Mathematical physics (mathematical physics)

7. Topology / analysis situs (topology)

8. Number theory (number theory)

9. Algebraic geometry (algebraic geometry)

As for astronomy and physics, Poincare's achievements include the introduction of chaos theory, whose contribution to classical mechanics, fluid mechanics, electromagnetism and optics may not be very significant, but its contribution to the establishment of quantum mechanics and relativity has indeed laid the foundation.

After Poincare, the French Academy of Sciences published an 11-volume collection of essays for him (uvres Publi é es sous les auspices de l'Acad é mie des Sciences), specifically:

Tome 1, É quations des Differentielles (differential equation)

Tome 2, Fonctions Fuchsiennes (Fox function)

Tome 3, É quations des Differentielles, Th é orie des fonctions (differential equation, function theory)

Tome 4, Th é orie des fonctions (function theory)

Tome 5, Alg è bre, Arithm é tique (Algebra, arithmetic)

Tome 6, G é m é trie, Analysis situs (Geometry, Topology)

Tome 7, Principles de m é canique analytique, Probl è me des trois corps (principles of analytical mechanics, three-body problem)

Tome 8, M é canique c é leste, Astronomie (astromechanics, astronomy)

Tomes 9-10, Physique Math é matique (mathematical physics)

Tome 11, M é moires divers-livre du centenaire (various commemorative articles, centenary commemorative essays)

In addition, Poincare also has a large number of handouts on various mathematics and physics courses, such as the three volumes of "Astromechanics handouts".

Poincare is also a popularizer of mathematics and physics and has written many bibliographies for the public. Some of Poincare's works are listed below:

1. Sur les é t é s des fonctions d é finies par les é quations aux differences é rences partielles: premiere these (on the properties of functions defined by partial differential equations), Gauthier-Villars (1879).

2. Les m é thodes nouvelles de la m é canique celeste (a new method of astromechanics), Gauthier-Villars, Tome 1,1892 Tome 2, 1893 X Tome 3, 1899.

3. La Science et l'Hypoth è se (Science and hypothesis), Flammarion (1902).

4. La Valeur de la Science (value of science), Flammarion (1905).

5. Cours d'astronomie G é n é rale (General Astronomy course), É cole polytechnique (1907).

6. Science et M é thode (Science and methods), Flammarion (1908).

7. Savants et é crivains (scholar and writer), Flammarion (1910).

8. Ce que disent les choses (so said), Hachette (1911).

9. Derni è res Pens é es (Last thought), Flammarion (1913).

In view of Poincare's academic status and influence, most of his works are available in many languages, such as Science and methods, Science and hypotheses and so on.

By the way, like Euler, Poincare has a bad look in his eyes.

3. Poincare's mathematical achievements Poincare was first of all a professional mathematician who dabbled in almost all fields of mathematics. In fact, there is no field in mathematics. I remember that Hilbert said that mathematics is not divided into majors, only branches and not. An attempt is made to introduce Poincare's mathematical achievements beyond the scope of this book, especially beyond the author's ability. Here are two famous examples for a brief introduction.

3A) Poincare disk. Poincare disk is a hyperbolic geometric model proposed by Bertrami (Eugenio Beltrami,1835-1900) but became famous by Poincare. To examine a disk on a plane, expressed in the plural, is a set.

Figure 2. The arc of a circle orthogonal to the boundary of the Poincare disk is a straight line on the Poincare disk. The study of the Poincare disk has inspired mathematicians and people other than mathematicians to think about geometry. Inspired by the Poincare disk problem, the Dutch printmaker M.C. Escher,1898-1972 created four prints called "the limit of Circle" in 1958, which is an excellent artistic embodiment of the Poincare disc layout problem (figure 3).

Figure 3. Poincare disc. The middle picture shows the third of Escher's engraving "the limit of circles", and the picture on the right shows the 6-4-2 triangular arrangement of Poincare disk 3b) Poincare conjecture. Mathematicians like Poincare think far beyond the problems they can solve. In 1900, Poincare, who studied topology, put forward the following conjecture: "any simply connected and closed 3-manifold is homeomorphic to a 3-sphere, that is, from a topological point of view, they are the same." The so-called 3-ball S3 is the ball defined by x2sphere y2sphere z2 + w ^ 2 = R2. The three-ball has a trivial basic group (trivial fundamental group), that is, any ring on it can contract to a point. Interestingly, the high-dimensional generalization of the Poincare conjecture was proved before the original conjecture was proved, while the Poincare conjecture itself was not proved by the Russian mathematician Gregori Perelman,1967- in three articles in 2002 and 2003, and was not approved by peer review in 2006. Perelman refused to give him the Fields Prize and the Clay Institute of Mathematics Award, which is particularly admired by the author. Poincare conjecture is one of the seven millennium mathematical problems, and its significance is not for the author to decide. For interested readers, please refer to the professional review.

4. Poincare's achievements in Mathematical Physics Poincare is a mathematician. Naturally, for physics, he first pays attention to astrophysics, which is the origin of physics. There is a strict analytical solution to the two-body problem through the interaction of gravity. Naturally, people want to extend this problem to the three-body problem or even the n-body problem (n is a small natural number). In 1887, King Oscar II of Sweden offered a reward for the solution to the three-body problem, and the prize was awarded to Poincare. Poincare did not solve the three-body problem, and even his paper contains many mistakes, but Poincare's paper ushered in a new era of astrophysics. Poincare put forward the concept of chaotic motion for the first time in his thesis. Chaos refers to a kind of dynamic behavior which is extremely sensitive to initial conditions. Nowadays, mixing theory has become an important branch of mathematics, and has penetrated into many disciplines such as physics, chemistry, sociology and so on.

Poincare's great contribution to physics is reflected in that he is the extraordinary founder of quantum mechanics and the theory of relativity.

4a). Quantum mechanics. Poincare's important contribution to quantum mechanics is that he proved in 1912 that the quantization of energy is a necessary and sufficient condition for obtaining Planck's blackbody radiation formula. Poincare's work puts an end to the efforts of physicists to understand (get rid of) the quantum concept since 1900, when Planck quantized the energy of light at a certain frequency to an integer multiple of v. In fact, Planck has been trying to prove that energy quantization is unnecessary, if not wrong, and has even come up with important concepts such as zero point energy. It was not until Poincare's mathematics was proved that Planck stopped, rather than the experimental results that Einstein explained the photoelectric effect by energy quantization in 1905, as reported in the general quantum mechanics literature. Poincare's work has not been mentioned in many textbooks of quantum mechanics. The author reiterates once again that from the perspective of theoretical rigor, Poincare's argument is indispensable, otherwise the quantization of energy has always been an unreassuring hypothesis. This proof is an impossible task for people with mathematical skills like Planck and Einstein. From a practical point of view, it is a bridge to quantum statistics and solid quantum theory, and it is easier to understand quantum statistics after understanding this truth. Einstein, Ellen Fester and others quickly developed solid quantum theory on the basis of Poincare's work.

Poincare began to think about whether Planck's formula could be obtained without introducing quantum discontinuities [Henri Poincar é, Sur la theorie de quanta, J. Phys.2, 5-34 (1912)]. He found that the conclusion was negative. Poincare analyzed the energy distribution between the oscillator and the atomic motion. The relationship between the average energy of the oscillator and the energy density of radiation is based on the stochastic phase approximation (Random phase approximation). Starting from the Boltzmann distribution, if the volume element of the phase space is dV, then the probability of the state in this space is e-E/kTdV, which is the basic principle of statistics. In other words, it can be expressed as the probability in the energy interval, dW=Ce-E/kT ω (E) dE, where the defined state density function ω (E) = dV / dE, which is the derivative of the phase space volume V contained in energy E. The only weight function that Poincare studies the average energy compatibility of the function form is ∞. Planck quantization is a necessary and sufficient condition for Planck distribution formula. This result cannot be obtained without a deep knowledge of statistical mechanics and mathematics. Poincare's work shows a leap of thinking that people feel confused. In fact, in the knowledge of him, there is no thinking jump. We feel the jump because we know less.

Poincare's contributions to the theory of relativity and quantum mechanics are both fundamental and decisive. His determination of the quantized condition as a sufficient and necessary condition for the blackbody radiation formula is no less than emphasizing the significance of Lorentz transformation to the special theory of relativity. This point has been ignored for a long time in the physics literature. The author is proud to be the first to realize this.

4B). Relativity. Poincare is very clear about the system of classical mechanics. As a mathematician, he should understand the transformation invariance of Euler's research equation in a short time, although the author has not seen any specific words in which Poincare talks about relativisim. Poincare expressed the principle of relativity as the principle that all physical phenomena should follow. Comparable to this, Curie (Pierre Curie,1859-1906) promoted symmetry to the object of physics. In addition, as a member of the Paris length (Standard) Bureau, the calibration of clocks, especially clocks that move with each other, is a problem he has been thinking about for a long time. In the article "time Measurement" in 1898, he pointed out that time has only the meaning of convention.

Poincare studied the invariant transformation of Jean x2+y2-z2=1 as early as 1881, which is actually the hyperbolic geometry of (2pl)-dimensional space, while special relativity, mathematically speaking, is only the hyperbolic geometry of (3pl)-dimensional space. Poincare's key contribution to special relativity is that he believes that Lorentz transformations should form groups, which finally finalizes the form of Lorentz transformations. The group formed by Lorentz transformation is called Lorentz group, while the larger space-time transformation group including translation is called Poincare group. The mathematics and physics of relativity, in the Poincare group. Special relativity is attributed to Einstein because Einstein derived a differential equation from the problem of time synchronization between moving clocks, and the solution of this equation is Lorentz transformation.

Poincare died in 1912. General relativity was constructed at the end of 1915, but the concept of gravitational wave, which was talked about in the later period of general relativity, first appeared in Poincare's 1905 paper (onde gravitique). Accelerating the moving charge produces electromagnetic waves, and Poincare suggests that the accelerated mass may radiate gravitational waves by analogy. The author does not dare to accept this analogy easily. Electric charge is the existence of polarity, the world of positive and negative charge pursues local electric neutrality, the total zero charge distribution has electric dipole moment, accelerated charge radiation electromagnetic wave. The mass is non-polar, and there is no theoretical support for whether the accelerated mass will radiate gravitational waves or not without the argument of mass dipole moment (dipole). Einstein himself was embarrassed that the gravitational waves derived later were reluctantly cobbled together.

Poincare has always taught physics. The author flipped through his É lectricit é et optique (electricity and Optics) handout and found that there was a lot of reference in it. The French contribute a lot to electricity and optics, and many of the details, I mean the details in which learning is created, should be added to our textbooks.

5. The thinker and literati Poincare Poincare is a thinker. As a philosopher with mathematical and physical background, his view is contrary to Russell Russell (Bertrand Russell,1872-1970) and Frege (Gottlob Frege,1848-1925) that mathematics is a branch of logic. Poincare believes that intuition is the life of mathematics, and mathematics cannot be derived from logic because it is not analytical. Poincare's philosophy of science is called practical contingency. I think it has something to do with Poincare's strong physical background. As far as physics is concerned, axiomatization is the end of effort rather than the source of learning.

Poincare admires the important role Convention plays in physics, so he is considered a fan of conventionalism. He believes that Newton's first law (Galileo's law) is not empirical, but the framework assumption of mechanical agreement; the geometry of physical space is also agreed. The geometry of the physical field or physical quantities such as the temperature gradient (if such a problem is studied) can be changed, and the space can be described as non-Euclidean geometry, but it can also be described as an Euclidean space, but its metric varies with the distribution of temperature. Of course, people are still used to the space of Euclidean geometry.

Poincare has left many collections of essays, including Science and hypothesis, the value of Science, Science and method, Mathematics and Logic, Scholars and Writers, and so on. Science and method is a collection of his lectures at the French Psychological Society, in which the ideas of creation and invention are summarized as two levels at the spiritual level: the first is the random combination of possible solutions to the problem, and the other is critical evaluation, that is, choice. "Science and hypotheses" is a classic of the philosophy of science in the 20th century. The most impressive sentence for the author is "Le savant doit ordonner; on fait la science avec des faits comme une maison avec pierres; mais une accumulation de faits n'est plus une science qu'un tas de pierres n'est une maison. Building science from facts is like building a house with stones, but the accumulation of facts is not science, like a pile of stones is not a house." In the value of Science, Poincare points out that mathematics has three purposes: to provide tools for the study of nature (Elles doivent fournir un instrument pour é tude de la nature), that is, the purpose of physics, as well as philosophical and aesthetic purposes. The so-called philosophical purpose is to assist philosophy in deepening the concepts of number, space and time. With regard to the physical and aesthetic purposes of mathematics, Poincare believes that we cannot sacrifice either of them. These two purposes are inseparable, and the best way to achieve one is to aim at the other, at least not to let the other escape from view (ces deux buts sont ins é parables et le Meilleur moyen d'attendre l'un c'est de viser de l'autre, ou du moins de ne jamais le perdre de vue). Said Poincare.

There are many popular quotes in the book "the value of Science", such as "Les Math é maques m é ritent d' ê tre cultiv é es pour ells-m ê mes (mathematics itself is worth ploughing)"; "Aussi l'homme ne peut ê tre heureux par la science, mais aujourd'hui il peut bien moins encore ê tre heureux sans elle (human beings will not be happy because of science, but today they may not be happy without science)." "Il ne faut donc pas croire que les th é ories d é mod é es ont é t é st é Riles et vaines"; "Et puis, pour chercher la v é rit é, il faut ê tre indh é packs, tout à fait independent. ... si nous voulons ê tre forts, it faut que nous soyons unis (in order to seek truth, man should be independent, completely independent. If we want to be strong, we should be a "one". Young friends who are willing to be science might as well consider this last sentence carefully.

Poincare spent a lot of time in his life Vulgarization de science his results and other scientific knowledge. The French word Vulgarization de science can be translated as the popularization of science, or the vulgarization of science, or the vulgarization of science. I think it depends on what level of scholars the parties concerned will fall into. Poincare's scientific popularization works are still nutritious professional classics in the eyes of the author and other professional scientists. Poincare among scientists is like the Impressionist painter among artists. His writings express his thoughts, but they seem to be discussing with you, and the phrase "ce n'est pas tout" often pops up in the sentence. The author thinks that this sentence should be written in all our textbooks to tell readers, especially students, that the current expression is not all of the problem at all, let alone the whole of this subject.

Poincare is said to be an impatient man (C'é tait un homme impatient qui é crivait vite) because he writes so fast that he even won a biography called Poincare: the impatient Man. Writing fast, it is inevitable to scribble mistakes, turnips quickly unwashed mud is a feature of Poincare's article, which is where he has been criticized. However, this is also the common fault of creating the mind-he does not have time to do the work of first-class or below-class scholars. Pascal's no time to be brief, Poincare's lack of time to modify, are unwilling to spend their time on things other than creation. A person who is immersed in creation must be a person with rich heart and tasteless appearance. However, the expression of a scientific giant can sometimes be very playful. For example, when it comes to speaking a foreign language, Poincare says, "… parler les langues é trang è res, voyez-vous, c'est vouloir marcher lorsqu'on est boiteux (speaking a foreign language is like wanting to walk even if you are lame)" (see la m é canique nouvelle).

People will wonder what makes Poincare such a creative genius. The answer is talent, unforgettable and persistent commitment to scientific problems. In addition, he has the ability to work on one problem consciously and subconsciously on another. Poincare has the ability to go straight to the core of the problem, including the origin and details of the problem. When Poincare reads other people's articles, he goes straight to the results and then constructs his own argumentation process. When he visited Gottingen, Germany, the students there were willing to discuss problems with him. If he finds nothing new, his regular comment is "A quoi bon?" .

In the author's mind, Poincare is close to the existence of God, and Einstein is overshadowed by him. Poincare was first and foremost a professional mathematician, for all the fields of mathematics, including all the fields of physics at that time, he devoted himself to it, he was rich, and he opened up new fields. It is really inconceivable that a person can have so many such profound creations in such a broad field. From the very beginning, the author gave up the attempt to introduce him in a comprehensive way, and there was nothing I could do about it. For those who are interested in Poincare, please read his works or monographs on his people and achievements.

As a great scholar, there is no doubt that Poincare has an excellent memory. It is said that he is an insatiable reader and never forgets it (Poinka é tait un lecteur insatiable et qu'il m é morisait facilement ce qu'il lisait). Like Euler, Poincare suffered from vision problems for a long time in his life. However, because the inner eye knows the sky, poor vision can also do spatial imagination, and you can immerse yourself in the twists and turns of geometry and topology (Il ne dessinait pas tr è s bien, mais faisait preuve de beaucoup d'imagination spatiale gr â ce à une solide vision int é rieure, qui lui permettait de se plonger dans les m é andres de la g é m é trie et de la topologie. Poincar'e was an explorer and adventurer, but of the jungles, deserts, and mountains of the spirit. He made fantastic journeys, but all those adventures took place in his mind.) Poincare's mental arithmetic ability, by which I mean the frontier exploration of mathematics without moving a pen and paper, is rare. Personally, I think, never forget, see the sky, this ability is a gift, but we vulgar people might as well specialize in training, which is certainly beneficial. However, it is not necessary to understand mental arithmetic as a child's thing like integer addition and multiplication. For the training of professional mathematicians and physicists, blind chess training can be used as an introductory course, and those who can blindly push the Poincare conjecture can be allowed to graduate.

Poincare stands proudly in the world of scientists because of his extraordinary intelligence, erudition and great achievements. There are comments that Poincare as a scientist is in its own category (Henri Poinka é was in a class by himself). Such people, alone, continue to bring so much new knowledge to mankind. Research is a completely private way of life for him. From this point of view, the so-called "learning is probably a matter discussed by two or three heartless people in an old house in the wilderness" has suddenly exposed that the knowledge referred to by Mr. Qian Zhongshu is certainly not of universal significance as Poincare has. Even for vegetarian people, the most appropriate career for three people to get together is to fight landlords rather than to do learning. It is funny to think about the cutting-edge scientific issues by hoping to hold a meeting together.

reference

[1] Henri Poincar é, Sur le probel é me des trois corps et les é quations de la dynamique (on three-body problems and dynamic equations), Acta Mathematica 13,1-270 (1890).

June Barrow-Green, Poincare and the Three Body Problem, AMS (1997).

Eric Temple Bell, Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincar é, Touchstone (1986).

[4] A. Logunov, Henri Poinka é and Relativity Theory, Nauka (2005).

[5] William B. Ewald (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press (1996).

[6] Henri Poincar é, Les mathematiques et la logique (Mathematics and Logic), La revue de m é taphysique et morale, pp.815-835 (1905); pp.17-38294317 (1906).

[7] Ferdinand Verhulst,Henri Poincar é: Impatient Genius,Springer (2010).

[8] Jean-Marc Ginoux, Christian Gerini, Henri Poincar é: A Biography Through the Daily Papers, World scientific (2014).

Jeremy J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincar é, Birkh ä user (2000).

Ernest Lebon, Henri Poincar é: Biographie, Bibliographie Analytique des É crits (Poincare: analytical Philology of Archives and writings), Gauthier-Villars (1912).

[11] Poincar é, Henri, La mesur du temps (measurement of time), Revue de M é taphysique et de Morale 6 Magi 1-13 (1898).

Peter Galison, Einstein's Clocks and Poincar é's Maps: Empires of Time,W. W. Norton & Company (2003).

[13] Jean-Paul Auffray, Einstein et Poincar é: Sur les traces de la Relativa é (Einstein and Poincare: the footprint of the Theory of Relativity).

[14] Henri Poincar é, Sur la dynamique de l'electron, Comptes Rendues 140,1504-1508 (1905).

Henri Poincar é, Sur la dynamique de l'electron, Rend. Circ. Matem. Palermo 21,129-175 (1906).

[16] Henri Poincar é, Les G é m é tries non-euclidiennes, Revue g é n é rale des Sciences pures et qul é es 2,769-774 (1891).

This article is selected from "majestic for one".

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